Lecture notes on embedded contact homology

TL;DR

These notes provide a comprehensive introduction to embedded contact homology (ECH) of contact three-manifolds, covering foundational concepts, origins, and recent applications to symplectic embedding problems, with detailed explanations and new insights.

Contribution

The notes compile and expand on foundational notions of ECH, include unpublished remarks and examples, and review recent applications to symplectic embedding problems.

Findings

Introduction of basic notions of ECH

Discussion of the origins and unpublished remarks

Application of ECH to symplectic embedding problems

Abstract

These notes give an introduction to embedded contact homology (ECH) of contact three-manifolds, gathering together many basic notions which are scattered across a number of papers. We also discuss the origins of ECH, including various remarks and examples which have not been previously published. Finally, we review the recent application to four-dimensional symplectic embedding problems. This article is based on lectures given in Budapest and Munich in the summer of 2012, a series of accompanying blog postings at floerhomology.wordpress.com, and related lectures at UC Berkeley in Fall 2012. There is already a brief introduction to ECH in the 2010 ICM proceedings, but the present notes give much more background and detail.